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Intuitive understanding of the relationship between the elasticityof objects and kinematic patterns of collisions

Michele Vicovaro1,2 & Luigi Burigana1

Published online: 9 December 2015# The Psychonomic Society, Inc. 2015

Abstract Horizontal collisions have long been used as a toolfor exploring people’s intuitive understanding of elementaryphysical laws. Here, we explored intuitive understanding ofthe relationship between the kinematic patterns of collisionsand the elasticity of the colliding objects. In Experiment 1A,we manipulated the simulated materials of two virtually col-liding spheres and asked the participants to judge whether thesimulated collisions appeared Bnatural^ or Bunnatural.^ Wedid the same in Experiments 1B and 2, but asked the partici-pants to adjust the velocities until the collisions appeared to beBperfectly natural.^ In Experiment 3, we removed pictorialcues to the materials of the colliding spheres and asked theparticipants to rate the bounciness of the materials, in view ofthe kinematics of simulated collisions. Overall, the resultsshowed that observers intuitively understood that collisionsbetween more elastic objects subtend a higher coefficient ofrestitution than collisions between objects with lesser elastic-ity. The results also highlighted some discrepancies betweenthe intuitive and Newtonian physics of collisions. Observerswere somewhat insensitive to violations of the principle ofenergy conservation, and their responses were influenced byirrelevant kinematic features of the collisions, such as the col-lision type and precollision velocity. We discuss our experi-mental results in relation to salient theoretical perspectives onintuitive physics.

Keywords Intuitive physics . Collisions . Causal perception .

Information integration .Material properties

Intuitive physics concerns people’s intuitive understanding ofelementary laws of physics. One may expect that people’spredictions about the behavior of physical objects should bequite accurate, given their abundant everyday experiences.However, research in intuitive physics reveals that this is notalways the case. For instance, a significant portion of the par-ticipants in McCloskey, Caramazza, and Green’s (1980) studypredicted that a ball rolling inside a C-shaped tube wouldfollow a curved path instead of a physically correct straightpath when exiting from the tube. The origin of the discrepan-cies between intuitive and Newtonian physics constitutes animportant theoretical question for perceptual and cognitivescientists. Some scholars (e.g., Proffitt & Gilden, 1989) havemaintained that the cognitive system is inherently limited, andthus Bpeople make judgments about natural object motions onthe basis of only one parameter of information that is salient inthe event^ (p. 384). In contrast, other scholars (e.g., Anderson,1981, 1983) have argued that the cognitive system can inte-grate multiple sources of stimulus information; thus, in prin-ciple, it can deal with multidimensional mechanical events.Hecht (2001; Hecht & Bertamini, 2000) suggested that peo-ple’s interpretations of physical events are based on the Bex-ternalization^ of body dynamics, rather than on the mechani-cal laws of motion (see also Yates et al., 1988).

A fundamental topic of Newtonian physics is the relation-ship between force and motion. How does an object move inresponse to the application of a force? People’s intuitive un-derstanding of this relation has been explored in various phys-ical contexts, such as projectiles exiting from curvilinear-shaped tubes (Cooke & Breedin, 1994; Kaiser, Proffitt, &Anderson, 1985; McCloskey, 1983; McCloskey et al., 1980;

* Michele [email protected]

1 Department of General Psychology, University of Padua, Padua, Italy2 Dipartimento di Psicologia Generale, Università di Padova, via

Venezia 8, I-35131 Padova, Italy

Atten Percept Psychophys (2016) 78:618–635DOI 10.3758/s13414-015-1033-z

http://crossmark.crossref.org/dialog/?doi=10.3758/s13414-015-1033-z&domain=pdf

McCloskey & Kohl, 1983). objects falling after having beenreleased by moving carriers (Kaiser, Proffitt, & McCloskey,1985; Kaiser, Proffitt, Whelan, & Hecht, 1992; Krist, 2000;McCloskey, 1983; McCloskey, Washburn, & Felch, 1983).projectiles thrown by virtual human characters (Hecht &Bertamini, 2000; Vicovaro, Hoyet, Burigana, & O’Sullivan,2014), and two-body collisions (Kaiser & Proffitt, 1987;Michotte, 1963; Runeson, 1977/1983; White, 2007).

In this study, we explored the intuitive physics of horizon-tal collisions, which are physical events with the followingfour characteristics: (1) A collision involves two sphericalbodies (which we call A and B), the masses of which areuniformly distributed (e.g., billiard balls); (2) the spheres haveno spin; (3) the collision takes place in a frictionless, isolatedenvironment; and (4) the spheres move horizontally on theobserver’s frontal plane. Figures 1A and B depict two typesof horizontal collisions.

For horizontal collisions like those represented in Figs. 1Aand B, the following equations specify the postcollision ve-locities of the spheres (vA and vB) as a function of theirprecollision velocities (uA and uB), their masses (mA andmB), and the so-called Bcoefficient of restitution^ (C):

vA ¼ mAuA þ mBuB þ mBC uB−uAð Þ½ �= mA þ mBð Þ; ð1ÞvB ¼ mAuA þ mBuB þ mAC uA−uBð Þ½ �= mA þ mBð Þ: ð2Þ

These equations are derived from the law of momentumconservation (see Kittel, Knight, & Ruderman, 1973). Herewe presume that a positive velocity denotes a motion from leftto right, whereas a negative velocity denotes a motion in theopposite direction. The coefficient of restitution (C) is a keyconcept for our study. The following equation is implied bythe equations above and specifies C as a function of the pre-and postcollision velocities of A and B:

C ¼ vB−vAð Þ = uA−uBð Þ: ð3Þ

Newtonian mechanics implies that 0 ≤ C ≤ 1. First, C can-not be greater than 1 because, in that case, the kinetic energyafter the collision would be greater than the kinetic energybefore the collision, which would violate the principle of en-ergy conservation. Second, C cannot be smaller than 0 be-cause this would imply the interpenetration of the collidingobjects, which is clearly impossible for solid bodies.1 Thesebounds for C imply that the relative postcollision velocity ofthe spheres (vB − vA) must lie between 0 and the relativeprecollision velocity of the spheres (uA − uB).

Probably due to their simplicity, horizontal collisions havelong been used as a privileged case for exploring the degree ofconsistency between intuitive and Newtonian physics. Albert

Michotte was a pioneer in this field of research. In his mostcelebrated experiment (Michotte, 1963, pp. 19–20), observerswere presented with two small squares aligned horizontally(see Fig. 1A for a 3-D version of Michotte’s stimuli). At apoint in time, one square (A) started moving toward the other(B), which was initially stationary. Upon being contacted, Bstarted movingwith the same velocity asA, whereas A came toa stop. The vast majority of observers described this scene bysaying that A Blaunched^ or Bkicked^ B—that is, that the mo-tion of A had caused the motion of B. This phenomenon wascalled the launching effect. Michotte adopted a Gestalt-theoretic approach to the perception of causality. He showedthat the physical plausibility of stimulus collisions was neithernecessary nor sufficient for perceiving the launching effect,which instead proved to depend on specific laws of perceptualorganization. For instance, Michotte showed that thelaunching effect occurred even when the value ofC subtendedby the simulated collision was greater than 1—that is, whenthe collision implied the violation of the laws of energy con-servation. In line with Michotte’s view, various researchershave highlighted inconsistencies between visual impressionsof causality and the physical laws of collisions (e.g., Bae &Flombaum, 2011; Scholl & Nakayama, 2002; White, 2007;Yela, 1952). Furthermore, Kaiser and Proffitt (1987) showedthat when simulated off-center collisions imply large devia-tions from the physically correct kinematic parameters, theyare still judged to be Bnatural^ by observers most of the time(see also O’Sullivan, Dingliana, Giang, & Kaiser, 2003).

Dissenting from Michotte, some researchers have empha-sized the substantial consistency between physical laws andsubjective judgments of collisions. Runeson (1977/1983) ar-gued that the visual system is Battuned^ to the physics ofcollisions. His theoretical approach, which is referred to asthe Bkinematic specification of dynamics^ (Runeson &Frykholm, 1983). relies on the idea that observers can Bpickup^ invariant dynamic properties of colliding objects from thekinematic patterns of collisions. The consistency between sub-jective judgments of collisions and Newtonian mechanics wasalso emphasized by Twardy and Bingham (2002). in theirstudy, the participants were presented with virtual animationsdepicting a ball falling from high above that, at the end of thefall, bounced (collided) several times upon the ground. Theauthors manipulated the value ofC implied by the rebounds ofthe ball and asked the participants to rate the Bnaturalness^ ofthe ball’s motion. The results showed that the naturalness rat-ings sharply decreased when the simulated value of C wasgreater than 1. This suggests that people are highly sensitiveto violations of the principle of energy conservation. On thebasis of these insights, Sanborn, Mansinghka, and Griffiths(2013) suggested that subjective judgments of causality arebased on the assumption that the value of C subtended byhorizontal collisions may only vary between 0 and 1. In arecent study, De Sá Teixeira, Oliveira, and Duarte Silva

1 For example, let us presume uA > uB. If C < 0, then (vB − vA) < 0, whichimplies vA > vB. This would mean that object A overtakes object B in thecollision process.

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(2014) found that participants’ predicted outcomes of colli-sions in a simulated Newton’s cradle device subtended valuesofC centered around .3; they argued that this value might havebeen Binternalized in order to reflect knowledge on the naturalstatistics of our ecological environment^ (De Sá Teixeira et al.,2014, p. 499).

From Bimmaterial^ to Bmaterial^ colliding objects

With few exceptions, previous research on the visual percep-tion of collisions has been conducted using Bimmaterial^ col-liding objects as experimental stimuli (e.g., simple, colorless2-D shapes). Apart from technical reasons, this seems to bebecause researchers have implicitly or explicitly assumed thatperceptual and cognitive judgments of collisions are not influ-enced by the materials from which the colliding objects aremade. From a physical viewpoint, however, materialproperties exert a dramatic influence on objects’ motion. Onthe one hand, the materials of two colliding objects may affecttheir masses, which—in turn—do influence the objects’postcollision velocities (by Eqs. 1 and 2). On the other hand,the materials of objects may also affect their elasticity values,which—in turn—do influence the postcollision velocities. Forexample, for horizontal collisions such as those depicted inFig. 1, other things being equal, the relative postcollision ve-locity (vB − vA) would be much smaller in a collision betweenPlasticine spheres (low elasticity) than in a collision betweenwooden spheres (high elasticity).

For collisions from common experience, the elasticity ofthe colliding objects has a strong influence on the resultingkinematic pattern (see Barnes, 1958, p. 8). This influence canbe formalized in terms of parameter C (see Eqs. 1–3). In gen-eral, for any given pair of pre-collision velocities uA and uB,the smaller the elasticity of the objects, the smaller parameter

C, the smaller the relative postcollision velocity (vB − vA). Thiswas empirically demonstrated by Cross (1999). who showedthat collisions involving comparatively elastic objects, such assteel balls and tennis balls, are characterized by higher valuesof C, as compared with collisions involving comparativelyinelastic objects, such as Plasticine balls and baseballs.2

Our present study focused on people’s intuitive understand-ing of the relationship between the elasticity of objects andkinematic patterns of collisions. The participants in our exper-iments were shown simulated collisions in which two collid-ing spheres had the same simulated material and the sameapparent size; hence, equal implied mass. Thus, we could varythe elasticity of the colliding objects (by manipulating theirsimulated material) while keeping their relative mass fixed atzero (mA − mB = 0). In a recent study by Vicovaro andBurigana (2014). manipulations of the simulated materials ofthe colliding spheres were, instead, aimed at varying theirrelative masses. The results of that study showed that peoplehave a good qualitative understanding of the relationship be-tween the relative masses of the colliding objects and thekinematic patterns of collisions. Recently, manipulations ofsimulated materials have also been used for varying the im-plied masses of virtual objects in research on the visual per-ception of objects’ stability (Lupo & Barnett-Cowan, 2015).

Few studies have explored people’s intuitive understand-ings of the relationship between the elasticity of collidingobjects and the kinematic patterns of collisions. De Sá

Fig. 1 (A) Three frames of a Michottean collision. (B) Three frames of a head-on collision. The letters A and B and arrows are added to indicate whichobjects are moving in the three stages of the collision events

2 These examples highlight that the scientific concept of elasticity shouldnot be confused with the concept of elasticity meant in everyday lan-guage. In everyday language, an object is only meant to be Belastic^ if(besides other conditions) it is deformable—namely, it undergoes visibleshape modifications consequent to the application of comparatively smallforces. In contrast, the scientific concept of elasticity is independent ofdeformability, as is shown by the fact that physically elastic objects caneither be deformable (tennis balls) or nondeformable (steel balls).

620 Atten Percept Psychophys (2016) 78:618–635

Teixeira et al. (2014) asked the participants in their study topredict the outcomes of collisions occurring between pendu-lum spheres in a simulated Newton’s cradle device. They ma-nipulated the implied elasticity of the colliding spheres byvarying their simulated materials, and found that its effect onthe participants’ responses was quite small. Warren, Kim, andHusney (1987) presented study participants with virtual ani-mations depicting a ball falling from high that bounced(collided) several times upon the ground at the end of the fall.They manipulated the value of C implied by the ball’s re-bounds and showed that the participants could precisely esti-mate the Bbounciness^ (elasticity) of the ball on the basis ofthe kinematics of its motion. To our knowledge, this is theonly evidence so far supporting the hypothesis that peoplemay intuitively understand the relationship between the kine-matics of collisions and the elasticity of objects.

Michottean and head-on collisions

We presented the participants with two types of simulatedhorizontal collisions, named BMichottean^ and Bhead-on.^The former was extensively used by Michotte (1963) in hisseminal work on the visual perception of causality; it is char-acterized by the fact that sphere B is stationary during theprecollision phase (uB = 0; see Fig. 1A). If mA = mB, then forMichottean collisions, Eqs. 1 and 2 simplify as follows:

vA ¼ uA 1 −Cð Þ=2; ð4ÞvB ¼ uA 1 þ Cð Þ=2: ð5Þ

These equations show that, after the collision, both spheresmove in the same direction, which is the same as that of Abefore the collision. The equations also show that vA decreaseswith C and may vary between 0 (when C = 1) and .5uA (whenC = 0), whereas vB increases with C and may vary between.5uA (when C = 0) and uA (when C = 1). For instance, ifC = 0,after the collision both spheres move with the same velocity,which is .5uA. If instead C = 1, sphere A remains stationaryafter the collision, whereas sphere B moves with the samevelocity that A had before the collision.

Head-on collisions are characterized by the fact that thespheres move toward each other with equal absolute velocities(uA = −uB; see Fig. 1B) during the precollision phase. IfmA = mB, for head-on collisions, Eqs. 1 and 2 come down to

vA ¼ −uAC; ð6ÞvB ¼ uAC: ð7Þ

These equations mean that, after the collision, both spheresmove in opposite directions with equal absolute velocities,which increases asC increases (in particular, vA = vB = 0 whenC = 0, and vA = −vB = −uAwhen C = 1).

Experiment 1A

In Experiment 1A, we presented the participants with virtuallysimulated Michottean and head-on collisions and manipulatedthe simulated materials of the colliding spheres, which couldbe wood, polystyrene, or Plasticine. The size of the sphereswas kept constant throughout the experiment, and the twospheres involved in one collision always had the same simu-lated material. This implies that spheres A and B had the sameimplied mass in each simulated collision, allowing us to applyEqs. 4–7, which presumemA =mB. We created six experimen-tal conditions, resulting from a 2 Collision Type (Michottean,head-on) × 3 Simulated Material of the Spheres (wood, poly-styrene, Plasticine) factorial design. Across the six experimen-tal conditions, the precollision velocities of the spheres (uAand uB) were kept constant, and the C coefficient implicit ineach collision was systematically varied within the intervalfrom 0 to 2 by varying the postcollision velocities vA and vB(see Eqs. 4–7). Thus, both physically plausible (0 ≤C ≤ 1) andimplausible (C > 1) simulated collisions were presented to theparticipants. The participants were asked to judge whethereach simulated collision appeared Bnatural^ or Bunnatural.^

Using the method described in the Experimental Designsection, we determined the most natural coefficient ofrestitution (NC)—defined as the implicit value of C in colli-sions with the greatest probability of appearing Bnatural^—foreach participant and experimental condition. By the methoddescribed in the Appendix, we also determined the physicalcoefficient of restitution (PC)—defined as the implied coeffi-cient of restitution by collisions between physical objects ofreal wood, polystyrene, or Plasticine—for each of the six ex-perimental conditions. We found that the PC was independentof the collision type (i.e., it was the same for Michottean andhead-on collisions) and varied with the spheres’material. Thevalues of PC were .66 for collisions between wood spheres,.61 for collisions between polystyrene spheres, and .13 forcollisions between Plasticine spheres. These measures showthat wood and polystyrene spheres have similar elasticity andare both more elastic than Plasticine spheres.

The comparison between the subjective measure NC andthe physical measure PC is a key step in testing the experi-ment’s main hypothesis—that is, that people have a good in-tuitive understanding of the relationship between the object’selasticity and the kinematic patterns of the collisions. If thiswere true, then the NC should be greater when the simulatedmaterial of the spheres is wood (PC = .66) or polystyrene(PC = .61) than when it is Plasticine (PC = .13), and shouldbe independent of the collision type (Michottean, head-on).We note two reasons why wood, polystyrene, and Plasticineare suitable materials for testing this hypothesis. One reason isthat they are common in everyday life, and thus it is likely thatthe participants may have good intuitive knowledge of theirmaterial properties. The other reason is that they allowed us to


test whether the participants’ naturalness judgments wereBcorrectly^ based on the material properties of the collidingspheres (e.g., elasticity), rather than on their masses.3

Another hypothesis to be tested by our experiment relatedto the participants’ sensitivity to violations of the principle ofenergy conservation. If the participants were sensitive to suchviolations (as was argued by Twardy & Bingham, 2002). thensimulated collisions subtending values of C greater than 1should appear decidedly Bunnatural.^

Method

Participants Twenty psychology students at the University ofPadua (ages 20–38 years; 13 females, seven males) participat-ed in the experiment on a voluntary basis. They all had normalor corrected-to-normal visual abilities. On average, they hadstudied physics at school for 2.8 years (SD = 1.64 years).

Stimuli and apparatus The stimuli were presented on a per-sonal computer equipped with a 37.5 × 30 cm CRTscreen anda keyboard. The participants sat at a distance of about 50 cmfrom the screen, the background of which was black. Twosimulated spheres of equal size were presented at the middleheight of the screen, with their centers aligned horizontally.Their apparent size, computed from the diameters of theircorresponding images on the screen (2.52 cm), was 8.4 cm3,subtending a visual angle of about 2.88°. For the Michotteancollisions (see Fig. 1A), at the beginning of each animation,one sphere (A) appeared close to the left edge of the screen andthe other sphere (B) appeared in the center. Then, 170 ms afterthe appearance of the spheres, A began to move horizontallyfrom left to right toward B, uniformly and without rotation,until it made contact with B. The postcollision velocities of thespheres in a trial were computed using Eqs. 4 and 5, afterinserting into them the value of C specified (by the experi-mental plan) for that trial. For head-on collisions (see Fig. 1B),at the beginning of each animation, one sphere (A) appearedclose to the left edge of the screen and the other sphere (B)appeared close to the right edge. Then, 170 ms after theirappearance, both spheres started moving simultaneously withopposite velocities until they made contact with each other inthe center of the screen. The postcollision velocities werecomputed using Eqs. 6 and 7. Each animation lasted 2,150 ms (the postcollision phase lasted 1,000 ms). Theprecollision velocities of the spheres were kept the samethroughout the experiment (A = 8.6 cm s−1, B = 0 cm s−1 for

Michottean collisions; A = 8.6 cm s−1, B = −8.6 cm s−1 forhead-on collisions).

We jointly manipulated the simulated materials of thespheres involved in the same simulated collision; these couldbe wood, polystyrene, or Plasticine. The spheres were createdwith 3D Studio Max. Photographic textures depicting the sim-ulated materials were attached to the spheres’ surfaces, and theirreflectances were regulated in order to increase the similaritywith their corresponding real spheres (see the next paragraph).The spheres thus created are depicted in Fig. 2. In each of thesix experimental conditions (2 Collision Types × 3 SimulatedMaterials), we varied the value of C underlying the collisions,as was required by the method of Brandomly interleaved stair-cases^ described in the Experimental Design section below.

The participants were allowed to touch and grasp realspheres made of wood, polystyrene, and Plasticine beforeand during the experiment. The masses of these spheres were55, 5, and 80 g, respectively, and their diameters were 4.5 cm.The real Plasticine sphere had a bright orange color similar tothat of the corresponding simulated sphere (see Fig. 2). Thesereal spheres were meant to facilitate the identification of thecorresponding simulated materials of the virtual spheres andwere also used to compute the PC for each of the six experi-mental conditions (see the Appendix).

Procedure Prior to this and the following experiments, theparticipants read and signed informed consent forms thathad been approved by the local ethics committee(Department of General Psychology, University of Padua).The instructions on the screen informed the participants thatthey would be presented with two simulated colliding spheres,which could be made of any one of three materials: wood,polystyrene, or Plasticine. At that point, the participants wereallowed to touch the real spheres set besides the keyboard andwere told that the spheres shown on the screen would faith-fully represent those real spheres. The participants were askedto judge whether each presented collision was Bnatural^ orBunnatural.^ The instructions specified that a collision had tobe judged as Bnatural^ or Bunnatural^ depending on its phys-ical plausibility (or implausibility). The instructions furtherspecified that a collision had to be judged Bunnatural^ whenthe postcollision motion of one or both spheres appeared to betoo fast or too slow, compared with their precollision motion.In each trial, the participants were allowed to view the stimu-lus as many times as they wanted by pressing the spacebar onthe keyboard; when they felt ready to respond, they had topress BN^ for the Bnatural^ response or BZ^ for the Bunnatural^one. After reading the instructions, the participants were pre-sented with five randomly chosen stimuli to familiarize themwith the task.

Experimental design We determined an upper naturalnessbound and a lower naturalness bound for each participant and

3 We argue, in this regard, that although masses are generally salient inperception, they should not affect the kinematic patterns of collisions ifmA = mB. For instance, the kinematic pattern of a horizontal collisioninvolving two wood spheres is similar to that of a horizontal collisioninvolving two polystyrene spheres (PC = .66 and .61, respectively), eventhough the former are rather different in mass from the latter.


each experimental condition. The upper naturalness boundmay be interpreted as the value ofC abovewhich the collisionappeared Bunnatural^ to the participant more than 50 % of thetime, presumably because the postcollision velocities of oneor both spheres looked too fast relative to their precollisionvelocities. Symmetrically, the lower naturalness bound maybe interpreted as the value of C below which the collisionappeared Bunnatural^ to the participant more than 50 % ofthe time, presumably because the postcollision velocities ofone or both spheres looked too slow relative to theirprecollision velocities. The naturalness intervalwas the inter-val of values of C that lay between the two bounds and gaverise to the impression of a Bnatural^ collision more than 50 %of the time. Finally, we defined the most natural coefficient ofrestitution (NC) as the midpoint of the naturalness interval.Assuming that the Bnaturalness^ judgments were normallydistributed, NC was tantamount to the value of C implicit ina collision with the greatest probability of appearing Bnatural.^

We used the method of Brandomly interleaved staircases^to estimate the individual upper and lower naturalness bounds(Levitt, 1971). In each of the six experimental conditions, wevaried the implicit value ofC in the collision, which could takeon 21 possible values, ranging from 0 to 2 in steps of 0.1.Figure 3 depicts an example of the randomly interleaved stair-cases procedure. The precollision velocities of the sphereswere fixed (see the Stimuli and Apparatus section above),whereas their postcollision velocities were varied as a functionof C, by applying Eqs. 4 and 5 for Michottean collisions, andEqs. 6 and 7 for head-on collisions.

Both naturalness bounds were estimated by generating twostaircases. For estimating the individual upper naturalnessbounds, one staircase started from C = 1. Every time the par-ticipant responded by judging the collision Bnatural,^ C wasincreased by one step; every time the participant respondedby judging the collision Bunnatural,^ C was decreased by onestep. Thus, the staircase changed its direction whenever theparticipant changed his or her answer, and it continued in thatdirection until the participant changed his or her answer again.Symmetrically, the other staircase started fromC = 2. The valueof C was decreased as long as the participant chose the Bunnat-ural^ response, and the staircase changed its direction wheneverthe participant changed his or her response. A Brun^ in a

staircase was a sequence of steps between two consecutivechanges in direction. Both staircases were terminated after eightruns.4 The individual upper naturalness bounds were estimatedby averaging the values of C at the midpoints of the last fourruns of both staircases (Levitt, 1971, p. 470).

We applied the same procedures to estimate the individuallower naturalness bounds, with one staircase starting fromC = 0 and the other from C = 1. Each staircase was increasedby one step after an Bunnatural^ response and decreased byone step after a Bnatural^ response. Both staircases were ter-minated after eight runs, and the final (averaging) computa-tions were the same as those described above.

Individual upper and lower naturalness bounds were esti-mated in each of the six (2 Collision Types × 3 SimulatedMaterials) experimental conditions. The experiment was di-vided into two blocks: Half of the participants received the 12staircases (3 Simulated Materials × 2 Bounds × 2 Staircases)relative to Michottean collisions first, whereas the other halfreceived the 12 staircases relative to head-on collisions first.The 12 staircases in each block were randomly interleaved toavoid anticipatory effects. The participants were allowed torest as much as they wanted after the first block. The experi-mental session could last from 25 to 35 min.

Results and discussion

Figure 4 shows the naturalness intervals (averaged across par-ticipants) obtained in the six conditions of the experiment. Wemarked four points for each interval: the mean upper natural-ness bound (top of the interval), the mean lower naturalnessbound (bottom of the interval), the most natural coefficient ofrestitution (NC; midpoint of the interval, thick line), and thephysical coefficient of restitution (PC, dashed line). The loca-tion of the naturalness intervals along the C continuum washigher when the simulated material of the colliding sphereswas wood or polystyrene than when it was Plasticine.Moreover, the location of the naturalness intervals on the Ccontinuumwas higher forMichottean collisions than for head-

Fig. 2 Simulated spheres used as stimuli in Experiments 1A and 1B. The simulated materials are, from left to right, wood, polystyrene, and Plasticine.To view this figure in color, please see the online issue of the journal

4 Because of the adaptive nature of the psychophysical method used inExperiment 1A, the number of trials for each participant and each stair-case varied.


on collisions, but this difference decreased in passing fromwood to polystyrene to Plasticine.

We performed a two-way within-participants analysis ofvariance (ANOVA; with the factors Collision Type andSimulated Material) on the NC measures to test this evidencestatistically. The main effects of both factors were significant:F(1, 19) = 14.69, p < .01, ηp

2 = .44, and F(2, 38) = 19.95,p < .001, ηp

2 = .51, respectively. Their interaction effects weremarginally significant, F(2, 38) = 3.06, p = .058, ηp

2 = .14.Bonferroni post-hoc comparisons (see Table 1) showed that,for both Michottean and head-on collisions, there was no sig-nificant difference between the NCs for wood and polysty-rene, which were both significantly greater than the NC forPlasticine. These comparisons also showed that when the sim-ulated material was wood, but not when it was polystyrene orPlasticine, the NC for the Michottean collision was larger thanthe NC for the head-on collision.

These results reveal that the participants intuitively under-stood that collisions between somewhat elastic objects such aswood or polystyrene spheres imply higher values of C com-pared with collisions between less elastic objects such as

Plasticine spheres. This finding supports the hypothesis thatpeople intuitively understand that the kinematic patterns ofcollisions are prominently influenced by the elasticity of thecolliding objects. Furthermore, the similarity between woodand polystyrene in the NCmeasure and their NCs over that ofPlasticine support the idea that the participants correctly basedtheir naturalness judgments on the material properties of thespheres (e.g., elasticity), rather than on their apparent masses.

Figure 4 also shows that the naturalness intervals of all sixexperimental conditions include values of C much higher thanthe PCs. The values of the NCs averaged over the participantswere 0.15–0.4 points higher than the corresponding PC values.In the case of Michottean collisions between wood or polysty-rene spheres, the collisions implying values of C much largerthan 1 were judged to be Bnatural^ by the participants most ofthe time, even though such collisions were at odds with theprinciple of energy conservation. This finding is inconsistentwith the claim that people are highly sensitive to violations ofenergy conservation laws (Twardy & Bingham, 2002). as wellas with the view that people may have Binternalized^ a physi-cally plausible value of C Bin order to reflect knowledge on the

Fig. 3 Full record of the responses of a hypothetical participant tested with randomly interleaved staircases. On the horizontal axis are the trials; on thevertical axis are the 21 possible values of C. The small squares and diamonds represent Bnatural^ and Bunnatural^ responses, respectively

Fig. 4 Mean naturalness intervals, mean NCs, and PCs for the six combinations of the Collision Type and Simulated Material of the Spheres factorsfrom Experiment 1A. The vertical bars represent the standard errors of the means


natural statistics of our ecological environment^ (De SáTeixeira et al., 2014, p. 499). Figure 4 also shows that, at var-iance with predictions from physics, the average NCs tended tobe higher for Michottean than for head-on collisions; such dif-ferences in NC reached statistical significance only for colli-sions between wood spheres (see Table 1). Overall, the resultsof our experiment suggest that people intuitively understandthat objects’materials affect the kinematic patterns of collisions,but people’s representations of the collision behavior of objectsare not fully consistent with physics.

Experiment 1B

In Experiment 1A, we used an Bindirect^method to obtain theNCmeasures: We first determined individual upper and lowernaturalness bounds by using randomly interleaved staircases,and then computed the NC as the arithmetic mean of bothbounds. In Experiment 1B, we obtained the NC using a Bdi-rect^ method—namely, a variant of the psychophysical meth-od of adjustment (Kingdom & Prins, 2010, pp. 48–51). In sixexperimental conditions, which were the same as those inExperiment 1A, the participants were asked to adjust the valueof C (by adjusting the postcollision velocities) until a simulat-ed collision appeared to be Bperfectly natural.^ Experiment 1Ballowed us to test whether the results of Experiment 1Awouldbe consistent across different psychophysical methods.

Method

Participants Twenty psychology students at the University ofPadua (ages 20–36 years; 13 female, sevenmales) participated

in the experiment on a voluntary basis. They all had normal orcorrected-to-normal visual abilities. None of them had partic-ipated in Experiment 1A. On average, they had studied phys-ics at school for 3.25 years (SD = 1.48 years).

Stimuli and apparatus The stimuli and apparatus were thesame as those of Experiment 1A.

Procedure The general instructions were the same as those inthe previous experiment. In addition, the participants were toldthat the difference between the postcollision velocities of thespheres would sometimes appear to be too large or too small ascompared to the difference between their precollision velocities,and that their task was to change the difference between thepostcollision velocities until the collision appeared to be per-fectly Bnatural^—that is, as similar as possible to how a realphysical collision should be. The participants were told that thedifference between the postcollision velocities could be in-creased or decreased by pressing BM^ or BZ^ on the keyboard,respectively, and that they had to press ENTER when the col-lision appeared to be perfectly Bnatural.^ In each trial, the par-ticipants were allowed to view the stimulus as many times asthey wanted by pressing the spacebar on the keyboard.

Experimental design The parameter C implicit in the simu-lated collisions could take on 21 different values, ranging from0 to 2 in steps of 0.1. In each of the six experimental condi-tions, individual NCs were obtained by generating four Bse-ries^ of trials. Two series started fromC = 2, and the other twostarted from C = 0. In each series, whenever the participantresponded by pressing BM^ on the keyboard, indicating thatthe difference between the spheres’ postcollision velocitieshad to be increased, C was increased by one step (which, byEqs. 4 and 5, or 6 and 7, implied increasing that difference).Conversely, C was decreased by one step whenever the par-ticipant responded by pressing BZ,^ indicating that the differ-ence between the postcollision velocities had to be decreased.Each series was terminated when the participant pressedENTER, indicating that the collision appeared perfectly Bnat-ural.^ Individual NCs were estimated by averaging the lastvalues of C in the four series (see note 4).

The experiment was divided into two blocks: Half of theparticipants received the 12 series (3 Simulated Materials × 4Series) relative to Michottean collisions first, and the other halfreceived the 12 series relative to head-on collisions first. The 12series in each block were presented in random order. The partic-ipants were allowed to rest as much as they wanted after the firstblock. The experimental session could last from 20 to 30 min.


Figure 5 shows the NC measures (averaged across partici-pants) obtained from each of the experiment’s six conditions.

Table 1 Results of the Bonferroni-corrected post-hoc pairedcomparisons for the six experimental conditions of Experiment 1A

(H, Wood) (M, Pol.) (H, Pol.) (M, Pla.) (H, Pla.)

(M, Wood) t = 3.955p < .001

t = 0.108p = .915

t = 1.764p = .094

t = 7.289p < .001

t = 8.032p < .001

(H, Wood) t = −2.510p = .021

t = −0.807p = .43

t = 3.770p = .001

t = 5.370p < .001

(M, Pol.) t = 2.333p = .031

t = 4.343p < .001

t = 5.492p < .001

(H, Pol.) t = −3.173p = .005

t = 4.257p < .001

(M, Pla.) t = 1.043p = .31

M and H stand for Michottean and head-on collisions, and Pol. and Pla.stand for polystyrene and Plasticine, respectively. For instance, (M, Pol.)stands for Michottean collisions with polystyrene spheres. Each t test had19 degrees of freedom, and the corrected value of α was .0033, whichresulted from dividing the original value of α (.05) by the number ofpaired comparisons (15). In this and the following tables, bold type standsfor statistically significant differences


The locations of NCs along the C continuum are similar tothose obtained from Experiment 1A, since they were higherwhen the simulated material of the colliding spheres waswood or polystyrene than when the material was Plasticine.Moreover, the NC is greater than the corresponding PC ineach of the six experimental conditions. Unlike the resultsfrom Experiment 1A, the locations of the NCs do not varywith collision type.

We performed a two-way within-participants ANOVA(with the factors Collision Type and Simulated Material) onthe individual NCmeasures. The main effects of the CollisionType factor were not significant, F(1, 19) = 0.80, p > .1,ηp

2 = .04, whereas the main effects of the SimulatedMaterial factor were significant, F(2, 38) = 26.52, p < .001,ηp

2 = .58. Their interaction effects were not significant,F(2, 38) = 1.90, p > .1, ηp

2 = .09. Bonferroni post-hoc com-parisons (see Table 2) showed no significant difference be-tween the NCs for wood and polystyrene, which were bothsignificantly greater than the NC for Plasticine.

Overall, the results of Experiment 1B confirmed the mainoutcomes of Experiment 1A. The participants showed a goodqualitative understanding that collisions between somewhatelastic objects (such as wood or polystyrene spheres) implyhigher values of C than collisions between less elastic objects(such as Plasticine spheres). Moreover, the values of the NCs

averaged across participants were 0.3 to 0.5 points higher thanthe corresponding values of the PCs. Thus, the main featuresof our experimental results proved consistent across differentpsychophysical methods.

Experiment 2

The PCmeasures that we obtained by the method described inthe Appendix indicated that the collision type (Michottean,head-on) did not affect the value of C implicit in physicalcollisions. By contrast, the results of Experiment 1A showedthat the NCmeasures tended to be higher for Michottean thanfor head-on collisions. This result was not replicated inExperiment 1B. Barnes (1958, pp. 7–8) pointed out that, be-sides the collision type, the objects’ precollision velocities arealso physically irrelevant to the value of C, at least when thevelocity varies within a relatively narrow range of values (onthe order of 0–10 cm s–1). The possible effects of this variableon the NC measures were not tested in Experiments 1A and1B, because the precollision velocity was kept constantthroughout both experiments. In Experiment 2, we obtainedthe NC measures by a psychophysical method similar to thatused in Experiment 1B, but we varied both the collision type(Michottean, head-on) and the precollision velocity of the col-liding objects (fast, slow) according to a factorial design. If theparticipants’ judgments of the naturalness of collisions wereconsistent with the Newtonian rules of collisions, then the NCmeasures should not vary with the collision type or theprecollision velocity, because these variables are irrelevant tothe value of C implicit in collisions.

In Experiment 2, we also used a larger sample of simulatedmaterials for the colliding objects than had been used inExperiments 1A and 1B. This approach was meant to testthe robustness and generalizability of the main outcomes ofthose experiments (i.e., a consistent association between theelasticity of objects and the kinematic patterns of collisions).

Fig. 5 Mean NCs and PCs for the six combinations of the Collision Type and Simulated Material of the Spheres factors from Experiment 1B. Thevertical bars represent the standard errors of the means

Table 2 Results of the Bonferroni-corrected post-hoc pairedcomparisons for the three levels of the Simulated Material factor inExperiment 1B

Polystyrene Plasticine

Wood t = 1.863 p = .07 t = 11.137p < .001

Polystyrene t = 5.944p < .001

Each t test had 39 degrees of freedom, and the corrected value of α was.0167 (i.e., .05/3)


The simulated colliding objects could be super balls (PC =.94), table tennis balls (PC = .80), tennis balls (PC = .71),rubber balls (from a beach rackets set, PC = .37), terracottaspheres (PC = .28), and Plasticine spheres (PC = .13). Theseobjects were chosen because they are somewhat common ineveryday life and because their PC measures are quite uni-formly distributed over the range of possible values of theparameter C.

We created 24 experimental conditions, resulting from a2 Collision Type (Michottean, head-on) × 2 PrecollisionVelocity (fast, slow) × 6 Simulated Colliding Object (superballs, table tennis balls, tennis balls, rubber balls, terracottaspheres, Plasticine spheres) factorial design. The twospheres involved in one collision always had the same sim-ulated material and size. In each of the 24 experimentalconditions, the NC measure was obtained by a method sim-ilar to that used in Experiment 1B (i.e., a variant of theadjustment method).

Method

Participants Twenty psychology students at the Universityof Padua (ages 19–29 years; 15 females, five males) partic-ipated in the experiment on a voluntary basis. They all hadnormal or corrected-to-normal visual abilities, and none ofthem had participated in Experiment 1A or 1B. On average,they had studied physics at school for 2.9 years(SD = 1.32 years).

Stimuli and apparatus The stimuli and apparatus were thesame as in Experiments 1A and 1B, except for the followingaspects. The simulated colliding objects varied not only inphotographic texture and reflectance, but also in size.Simulated tennis balls had the largest size (11.74 cm3), sincetheir diameter measured on the screen was 2.82 cm,subtending a visual angle of 3.23°. The size of the super balls,terracotta spheres, and Plasticine spheres was 8.4 cm3 (diam-eter = 2.52 cm, visual angle = 2.88°). Finally, the size of thetable tennis and rubber balls was 4.34 cm3 (diameter =2.22 cm, visual angle = 2.54°). The spheres thus created aredepicted in Fig. 6. The size, photographic texture, and reflec-tance of the simulated objects were meant to increase therealism of their appearance and the similarity with their cor-responding real objects (see the next paragraph). The

precollision velocity of sphere A could be either 6.7 or11.9 cm s−1 in both the Michottean and head-on collisions.The precollision velocity of B was null in the Michotteancollisions, whereas it could be either −6.7 or −11.9 cm s−1 inthe head-on collisions.

As in Experiments 1A and 1B, the real objects weremeant to facilitate the identification of the correspondingsimulated colliding objects, and they were also used tocompute the PC for each of the 24 experimental conditions(see the Appendix). The diameter of the real tennis ballswas 6 cm (mass = 58 g); that of the super balls, terracottaspheres, and Plasticine spheres was 4.5 cm (masses = 40,105, and 80 g, respectively); and the diameter of table ten-nis and rubber balls was 3.5 cm (masses = 2 and 12 g,respectively).

Procedure The procedure was the same as that of Experiment1B (i.e., a variant of the method of adjustment).

Experimental design The experimental design was the sameas that of Experiment 1B, except that in each of the 24 exper-imental conditions, individual NCs were obtained by generat-ing two Bseries^ of trials rather than four. One series startedfrom C = 2, and the other started from C = 0. With respect toExperiment 1B, we decreased the number of Bseries^ in orderto maintain the experiment at a reasonable length, given theincreased number of experimental conditions. The experimentwas divided into two blocks: Half of the participants receivedthe 24 series (2 Series × 2 Precollision Velocities × 6Simulated Colliding Objects) for Michottean collisions first,and the other half received the 24 series for head-on collisionsfirst. The 24 series in each block were presented in randomorder. The participants were allowed to rest as much as theywanted after the first block. The experimental session couldlast from 30 to 40 min.


Figure 7 shows the NC measures (averaged across partici-pants) as a function of the PC measures for the six levels ofSimulated Colliding Objects factor (horizontal axis) and eachcombination of levels of the Collision Type and PrecollisionVelocity factors (separate curves). Overall, the locations of theNCs along the C continuum increased with the PCs of the six

Fig. 6 Simulated objects used as stimuli in Experiment 2. The simulated objects are, from left to right, a Plasticine sphere, terracotta sphere, rubber ball,tennis ball, table tennis ball, and super ball. To view this figure in color, please see the online issue of the journal


types of simulated colliding objects.5 Moreover, the verticalseparation of the curves revealed that the location of the NCswas also conditional on Collision Type and PrecollisionVelocity. Specifically, the NCs tended to increase in passingfrom head-on toMichottean collisions and from fast to slow inprecollision velocity.

We performed a three-way within-participants ANOVA(with the Collision Type, Precollision Velocity, andSimulated Colliding Object factors) on the individual NCmeasures. The main effects of the three factors were signifi-cant: F(1, 19) = 10.93, p < .001, ηp

2 = .37; F(1, 19) = 30.24,p < .001, ηp

2 = .61; and F(5, 95) = 54.11, p < .001, ηp2 = .74,

respectively. The effects of the two-factor Collision Type ×Precollision Velocity interaction were significant, F(1, 19) =8.26, p < .01, ηp

2 = .30. The effects of the two-factor interac-tions for Collision Type × Simulated Colliding Object andPrecollision Velocity × Simulated Colliding Object were notsignificant, F(5, 95) = 0.96, p > .1, ηp

2 = .05, and F(5, 95) =0.94, p > .1, ηp

2 = .05, respectively. Neither was the effect ofthe three-factor interaction significant, F(5, 95) = 1.36, p > .1,ηp

2 = .07.

Bonferroni post-hoc comparisons (see Table 3) showed thatthe NC measures for the six types of simulated colliding ob-jects, averaged across the Collision Type and PrecollisionVelocity factors, were all significantly different from one an-other. These results support the robustness and generalizabil-ity of the main outcomes of Experiments 1A and 1B—that is,that participants intuitively understood that collisions betweenmore elastic objects imply higher values of C than collisionsbetween less elastic objects.

Bonferroni-corrected post-hoc comparisons (see Table 4)also showed that the NC measure, averaged across the sixlevels of the Simulated Colliding Object factor, was signifi-cantly lower for the head-on collision with a fast precollisionvelocity than for the other experimental situations. Moreover,the NC measure was significantly lower for the Michotteancollision with a fast precollision velocity than for theMichottean collision with a slow precollision velocity.Overall, these results indicate that the NCs are also sensitiveto kinematic conditions that, in real physical collisions, haveno impact on restitution coefficient C.6 Thus, they are exam-ples of a discrepancy between intuitive and Newtonian

5 Figure 7 shows that there is no relationship between the size of thesimulated objects and the NC measures. As illustrative examples, wecan see that for some pairs of simulated objects with the same size (e.g.,terracotta spheres and super balls) the NCs were very different from oneanother. Symmetrically, for some pairs of simulated objects with differentsizes (e.g., tennis balls and table tennis balls), the NCs were quite similarto one another. We point out, however, that the possible influence of sizeon the NC measures would be better explored in an experiment in whichthe size of the colliding objects is varied independently of their simulatedmaterials.

Fig. 7 MeanNCmeasures fromExperiment 2, represented as a function of thePC of each type of simulated colliding object, for each combination of theCollision Type and Precollision Velocity factors. The vertical bars represent the standard errors of the means

6 Table 4 shows that the NC measure for the Michottean collision wassignificantly higher than that for the head-on collision only for Bfast^ levelof the Precollision Velocity factor. This suggests that the effects of theCollision Type factor may become evident only when the simulated col-lision is characterized by a relatively high precollision velocity. InExperiments 1A and 1B, the precollision velocity was intermediate be-tween the slow and fast levels of Experiment 2, and this may explain why,in those experiments, the effects of the Collision Type factor were some-what unclear (i.e., its main effects were statistically significant in Exp. 1A,but not in Exp. 1B).


physics of collisions, to which we shall return in the GeneralDiscussion section. We also note that, similar to what wefound in Experiments 1A and 1B, the values of the NCs (av-eraged across the Collision Type and Precollision Velocityfactors) were 0.2 to 0.5 points higher than the correspondingvalues of the PCs.

Experiment 3

In horizontal collisions, when both spheres have thesame size and material, a one-to-one relationship holdstrue between the kinematic pattern of a collision and theelasticity of the spheres. The relationship is specified bythe following equation:

Elasticity A; Bð Þ ¼ f Cð Þ ¼ f vB −vAð Þ= uA −uBð Þ½ �; ð8Þ

where f is a suitable increasing monotone function and theother symbols are defined as before. This equation from phys-ics shows that, in principle, it is possible to determine theelasticity of the colliding spheres when the kinematic patternof the collision is available.

Whereas in the previous experiments we tested the influ-ence of objects’ elasticity on the apparent Bnaturalness^ of thekinematic patterns of collisions, in Experiment 3 we adopted aconverse approach, by testing whether people can infer theelasticity of colliding objects from the kinematics of colli-sions. Hence, we removed the pictorial cues to the materialof the spheres, varied the implicit value of parameter C in thecollisions, and asked the participants to rate the bounciness ofthe spheres’ materials on the basis of the kinematics of thecollisions.7 Should the participants be able to infer the elastic-ity of colliding objects from the kinematic pattern, then therated bounciness of the spheres would increase as the implicit

value of coefficient C in the collisions increased. InExperiment 3, we also tested whether the rated bouncinesswas influenced by kinematic conditions that, in real collisions,are irrelevant to the value of C. We thus manipulated collisiontype and precollision velocity according to a 2 Collision Type(Michottean, head-on) × 2 Precollision Velocity (fast, slow)factorial design. The implicit value of parameter C in the sim-ulated collisions was varied for each combination of levelswithin the two experimental factors (see the ExperimentalDesign section below).

In a study similar to the experiment we are going to de-scribe, Warren et al. (1987) presented their observers withvirtual animations of a ball that fell from high and bounced(collided) several times upon the ground at the end of the fall.The observers were asked to rate the bounciness of the ball onthe basis of the kinematic information provided in the anima-tion. The results showed a correlation between the ratedbounciness of the ball and the value of C implied by thebounces. This suggests that people can Binfer^ the bouncinessof an object on the basis of the kinematics of its bounces.

The approach we adopted in this experiment also bearssome resemblance to that used in research on the visual per-ception of relative mass among colliding objects (Gilden &Proffitt, 1989, 1994; Runeson, 1977/1983, 1995; Runeson &

7 We asked the participants to rate Bbounciness^ rather than Belasticity^because the physical concept of elasticity is more akin to bounciness thanto elasticity, in everyday parlance (see also note 2).

Table 3 Results of the Bonferroni-corrected post-hoc paired comparisons for the six levels of the Simulated Colliding Object factor in Experiment 2

Terracotta Rubber Tennis TableTennis Super Ball

Plasticine t = −4.053p < .001

t = −10.21p < .001

t = −12.498p < .001

t = −16.396p < .001

t = −20.642p < .001

Terracotta t = −8.33p < .001

t = −10.632p < .001

t = −15.04p < .001

t = −17.51p < .001

Rubber t = −4.04p < .001

t = −7.564p < .001

t = −12.048p < .001

Tennis t = −3.509p < .001

t = −6.527p < .001

Tabletennis t = −3.721p < .001

Each t test had 79 degrees of freedom, and the corrected value of α was .0033 (i.e., .05/15)

Table 4 Results of the Bonferroni-corrected post-hoc pairedcomparisons for the four combinations of Collision Type andPrecollision Velocity factors in Experiment 2

(M, Fast) (H, Slow) (H, Fast)

(M, Slow) t = 2.803p = .006

t = 2.253p = .026

t = 8.056p < .001

(M, Fast) t = 0.189p = .85

t = 6.372p < .001

(H, Slow) t = 7.309p < .001

Each t test had 119 degrees of freedom, and the corrected value of α was.0083 (i.e., .05/6)


Vedeler, 1993; Todd &Warren, 1982). For experiments in thisarea, observers are typically presented with simulated colli-sions between abstract shapes and are asked to judge the rel-ative masses of the colliding objects. Observers are thus re-quired to judge a stable property of objects (relative mass) onthe basis of the kinematic information provided by simulatedcollisions.

Method

Participants Fifteen psychology students at the Universityof Padua (ages 20–36 years; seven females, eight males)participated in the experiment on a voluntary basis. Theyall had normal or corrected-to-normal visual abilities, andnone of them had participated in Experiment 1A or 1B. Onaverage, they had studied physics at school for 2.2 years(SD = 1.47 years).

Stimuli and apparatus The stimuli and apparatus were thesame as in Experiments 1A and 1B, except that the simulatedspheres (created by 3D Studio Max) all had a smooth, green-ish appearance. The spheres had a fixed size (8.4 cm3); theirdiameters measured on the screen was 2.52 cm, subtending avisual angle of 2.88°. The precollision velocity of sphere Acould be either 6.7 or 11.9 cm s−1 in both the Michottean andhead-on collisions. The precollision velocity of B was null inthe Michottean collisions, whereas it could be either −6.7 or−11.9 cm s−1 in the head-on collisions. No real spheres werepresented to the participants before or during the trials.

Procedure The instructions on the screen informed the par-ticipants that they would be presented with two simulatedcolliding spheres and that their task was to rate, by integernumbers from 0 to 100, the bounciness of the materials fromwhich the colliding spheres seemed to be made. The partici-pants were told that the materials were not visually discerniblebecause the spheres had been covered with a very thin layer ofgreenish paper, and that their task was to estimate thebounciness of the hidden material on the basis of the spheres’collision behavior. They were also told that both spheres ineach collision were made of the same material. The instruc-tions further specified that the spheres could be made of dif-ferent materials, such as Plasticine, iron, wood, polystyrene,rubber, and so forth, and that the participants had to respond 0if they believed that the material was not bouncy at all (such asPlasticine). If, instead, they believed that the material was verybouncy (such as that of super balls), they had to respond 100.Finally, if they believed that the material had an exactly inter-mediate bounciness between those extremes, they had to re-spond 50. In each trial, the participants were allowed to viewthe stimulus as many times as they wanted by pressing thespacebar on the keyboard; when they felt ready to respond,they had to press ENTER on the keyboard, type the chosen

number, and then press ENTER again. After reading the in-structions, the participants were presented with eight colli-sions that were representative of the entire sample of experi-mental stimuli, in order to familiarize them with the task.

Experimental design For each combination of levels for thefactors Collision Type (Michottean, head-on) and PrecollisionVelocity (fast, slow), we varied the value of coefficient C insix steps (values of 0, 0.2, 0.4, 0.6, 0.8, 1). Overall, the par-ticipants had 48 test trials, resulting from a 2 Collision Type ×2 Precollision Velocity × 6 Value of C Implicit in Collisions ×2 Repetition factorial design.


Figure 8A shows the rated bounciness of the spheres’ mate-rials (averaged across participants and repetitions) as a func-tion of the value ofC implicit in the collisions (horizontal axis)for each combination of levels for the Collision Type andPrecollision Velocity factors (separate curves). Overall, therated bounciness increased as the implicit value of C in thecollisions increased. The pattern of divergent curves revealedthat the rate at which the judged bounciness increased (slopeof the curves) was greater for head-on than for Michotteancollisions and greater for the fast than for the slow precollisionvelocity.

In order to test this evidence statistically, we performed athree-way within-participants ANOVA (with the Value of C,Collision Type, and Precollision Velocity factors) on thespheres’ rated bounciness values (averaged across repetitions).Themain effects of the three factors were significant: F(5, 70) =127.8, p < .001, ηp

2 = .9; F(1, 14) = 9.94, p < .01, ηp2 = .41; and

F(1, 14) = 62.47, p < .001, ηp2 = .82, respectively. The effects of

the two-factor interactions for Value of C × Collision Type andValue of C × Precollision Velocity were significant, F(5, 70) =22.54, p < .001, ηp

2 = .62, and F(5, 70) = 10.99, p < .001,ηp

2 = .44, respectively. The effects of the two-factor interactionbetween Collision Type and Precollision Velocity were not sig-nificant, F(1, 14) = 0.51, p > .1, ηp

2 = .03, nor was the effect ofthe three-factor interaction, as (5, 70) = 1.85, p > .1, ηp

2 = .12.Linear regression analyses (see Fig. 8A) showed that the rate

at which the judged bounciness increased as coefficient C in-creased was greatest for the fast head-on collision (slope =95.47, intercept = 4.56, R2 = .83), followed by the slow head-on collision (slope = 66.49, intercept = 3.02, R2 = .69), then bythe fast Michottean collision (slope = 45.11, intercept = 16.80,R2 = .29), and finally by the slowMichottean collision (slope =29.53, intercept = 11.09, R2 = .29). As is shown by Fig. 8A andthese statistical results, within each of the four combined levelsof the Collision Type and Precollision Velocity factors, thebounciness estimated by observers steadily increased (at differ-ent rates) as the value of coefficient C increased. This generaltrend supports the hypothesis originating our experiment,


because it illustrates that observers can consistently infer theelasticity of spheres’ materials on the basis of the kinematicpatterns of collisions.

It is worth noting, however, that the participants’ responseswere not fully consistent with what physical science wouldpredict. The effects of the Value of C factor were conditionalon the levels of the Collision Type and Precollision Velocityfactors. Specifically, for any fixed value of C, the ratedbounciness of the spheres tended to increase in passing fromthe Michottean to the head-on collision, and from the slow tothe fast precollision velocity. Notably, both collision type andprecollision velocity appear to have opposite effects on therated bounciness in this experiment and on the NC measuresof Experiment 2 (see Figs. 7 and 8A for a comparison). Wewill provide an explanation of this apparent contrast in theGeneral Discussion section. Below, we discuss a responsemodel of the rated bounciness of colliding spheres that mayaccount for the results of Experiment 3.

A response model of the rated bounciness of collidingspheres

The response model we considered is expressed by this equa-tion:

Rated Bounciness A;Bð Þ ¼ r � vB −vAð Þp= uA −uBð Þq; ð9Þ

where uA, uB, vA, and vB are the pre- and postcollision veloc-ities of spheres A and B in a collision, and p, q, and r are theparameters to be estimated. This equation is akin to Eq. 8above, but it is more Bflexible^ because it includes parametersp and q, which may be interpreted as measures of the separatecontributions of vB − vA and uA − uB in determining the ratedbounciness of the spheres (parameter r is a simple rescalingcoefficient). Specifically, if p = q (i.e., the relative post- andprecollision velocities contribute equally to rated bounciness),then the right-hand side of Eq. 9 would amount to a specialcase of physical Eq. 8, in which f(C) = r × Cp. In this case, the

rated bounciness would be independent of the Collision Typeand Precollision Velocity factors, and the curves correspond-ing to each combination of levels of the two factors wouldcoincide. Otherwise, if p ≠ q, the relative post- andprecollision velocities would contribute unequally to the ratedbounciness of the colliding spheres. This would imply a pat-tern of diverging curves in which the rate of variation of therated bounciness as a function of C would be dependent uponthe specific combination of the levels of the Collision Typeand Precollision Velocity factors.8

We fitted the response model of Eq. 9 (a power-ratio mod-el) to the data of Experiment 3, and obtained the followingestimates by the least-squares criterion: p = .669, q = 0.057,and r = 13.348. Figure 8b shows the curves implied by Eq. 9when these estimates are substituted for the parameters in theformula, and uA is set to 6.7 for slow and to 11.9 for fastprecollision velocity (recall that uB = 0 in Michottean colli-sions and uB = −uA in head-on collisions). We compared theperformance of this model with that of the linear model thatwas formed of the regression lines for the four combinationsof levels of the Collision Type and Precollision Velocity fac-tors. The linear model has two parameters (intercept andslope) for each regression line (i.e., eight parameters). Thestatistical results for that model have been listed above andare illustrated in Fig. 8A. The square roots of the residualvariance were 17.91 for the power-ratio model and 17.74 forthe linear model. Thus, the power-ratio model proves almostas good as the linear model, in terms of accuracy, while beingdecidedly simpler with respect to the number of parameters(three rather than eight).

Fig. 8 (a) Mean rated bounciness from Experiment 3, represented as afunction of the coefficient C, for each combination of the Collision Typeand Precollision Velocity factors. The vertical bars represent the standarderrors of the means. Regression lines (see the text) are also represented for

each combination of the two factors. (b) Predicted rated bounciness foreach combination of the Collision Type and Precollision Velocity factorsaccording to the power-ratio model of Eq. 9 (with p = .669, q = 0.057, andr = 13.348)

8 Recall that uB = 0 in Michottean collisions and uB = −uA in head-oncollisions; by virtue of Eq. 3, this implies that the right-hand side of Eq. 9becomes r × (uA)

p−q ×Cp for the former and r × (2uA)p−q ×Cp for the latter

type of collision. When, for instance, p > q, these equations imply that therated bounciness increases more withC in the case of a fast than of a slowprecollision velocity. Moreover, for a fixed precollision velocity (uA), therated bounciness increases more with C in the case of head-on collisionsthan in the case of Michottean collisions [because (2uA)

p−q > (uA)p−q].


The most salient feature of these results is that p = .669is clearly greater than q = 0.057. This is the algebraic rea-son for the kind of divergence visible in Fig. 8B (see note8). Within the frame of the response model of Eq. 9, thismeans that the relative postcollision velocity proved tocontribute much more to the rated bounciness of the col-liding spheres than the relative precollision velocity.Significantly, similar evidence was obtained in experi-ments on the visual perception of the relative masses ofcolliding objects (Gilden & Proffitt, 1989, 1994; Todd &Warren, 1982). A possible interpretation of this finding isthat the relative postcollision velocity was more salient tothe participants than the relative precollision velocity. Thismay be due to the fact that, at the time of the response, thememory trace for the postcollision phase was more recent(hence, more Bvivid^) than that for the precollision phase.This may have prompted the participant to base his or herresponse more on the relative postcollision velocity thanon the relative precollision velocity.

To sum up, the fact that rated bounciness increases forincreasing C is a sign of the consistency between intuitiveand Newtonian physics. In contrast, the fact that ratedbounciness depends on collision type and precollision veloc-ity is a sign of inconsistency between intuitive and Newtonianphysics, and supports the idea that the postcollision phase ismore salient than the precollision phase in the task of ratingthe bounciness of colliding spheres.

General discussion

From Paolo Bozzi’s (1959) seminal work onward, people’sintuitive understanding of the relationship between forceand motion has been explored using simulations of physicalevents involving abstract, Bimmaterial^ objects. We under-took our present study in light of the belief that a deeperunderstanding of the force–motion topic requires the use ofmore realistic physical simulations involving Bmaterial^stimuli.

On the whole, the results that we obtained support thehypothesis that people intuitively understand the relationshipbetween the elasticity of objects and the kinematic patterns ofcollisions. Specifically, the results of Experiments 1A, 1B, and2 showed that the implicit coefficient C in collisions that ap-peared Bnatural^ consistently varied with the simulated mate-rials of the colliding spheres. Moreover, the results ofExperiment 3 showed that the rated bounciness of the collid-ing spheres consistently depended on the implicit coefficientC in the collisions. These findings are in line with the results ofrecent studies (Lupo & Barnett-Cowan, 2015; Vicovaro &Burigana, 2014) showing that people properly take accountof material properties when asked to judge the physical be-havior of objects.

On the influences of collision type and precollision velocityon the judged naturalness of collisions

The results of our experiments also highlighted some discrep-ancies between intuitive and Newtonian physics of collisions.One discrepancy is the influence (i.e., the main and interactioneffects) of collision type and precollision velocity both on theparticipants’ judgments of the naturalness of collisions (Exp.2) and on their judgments of the bounciness of collidingspheres (Exp. 3). Above we provided a tentative explanationof the results of Experiment 3, in terms of a greater saliency ofthe postcollision phase on participants’ bounciness ratings.The explanation of the results of Experiment 2 that we arenow going to suggest is closely related to that conjecture.

Let us presume that when both the simulated material of thecolliding objects and the kinematic pattern of the collision areavailable, which is the case in Experiment 2, the participantinfers the elasticity of the objects from both of these sources ofstimulus information. We call the Belasticity from the materialcues^ that implied by the simulated material, and the Belastic-ity from the kinematic cues^ that implied by the kinematics ofthe collision.9 Let us also presume that a simulated collision isjudged as Bnatural^ or Bunnatural^ depending on the consis-tency or inconsistency between the elasticity from the materialcues and the elasticity from the kinematic cues. For instance,in a head-on collision between Plasticine spheres withC = (vB − vA) / (uA − uB) = 1, presumably the elasticity sug-gested by the material would appear much lower than thatsuggested by the kinematics, which would prompt an Bunnat-ural^ response. These assumptions caused us to conjecturethat, in Experiment 2, the participants adjusted the parameterC implicit in collisions until the elasticity from the kinematicsmatched the elasticity from the material.

Now, in Experiment 3, only the kinematic cues to elasticitywere in action and we saw that, for any fixed value of C, thefour experimental conditions formed the sequence(Michottean slow, Michottean fast, head-on slow, head-onfast) when ordered by the increasing rated bounciness (seeFig. 8). We interpreted this result as being attributed to thesaliency of the relative postcollision velocity. By contrast, inExperiment 2, the material cues and the kinematic cues wereboth in action and, following the interpretation in the preced-ing paragraph, the participants’ task was to adjust parameter Cso that a match between elasticity from the material (fixed)and elasticity from the kinematics was attained. In these con-ditions, and according to a basic compensation principle (if a+ b = c + d and a < c, then b > d), it is only natural to expectthat, for a fixed material, the order between the four

9 At the current stage of our research, we cannot draw any conclusionabout the psychological processes related to the participants’ judgmentsof Belasticity from material cues.^We can only speculate that both visualand haptic perceptual processes might be involved, as well as high-levelcognitive processes related to the participant’s past experiences.


experimental conditions found by increasing the adjusted val-ue of C would be the opposite of the order mentioned above(i.e., it would be head-on fast, head-on slow, Michottean fast,Michottean slow). Actually, with few exceptions, this was theorder revealed by the data of Experiment 2 (see Fig. 7).

To sum up, if our assumptions about the psychologicalprocesses underlying the experimental tasks are correct,then the (physically unexpected) influences of the collisiontype and precol l is ion veloci ty on the resul ts ofExperiments 2 and 3 allow for a unified explanation. Thisrelies on the hypothesis that the elasticity inferable fromthe kinematic pattern of a collision depends more stronglyon the postcollision than on the precollision phase of theevent.

On the violation of the principle of energy conservation

The results of our experiments also highlight another note-worthy discrepancy between the intuitive and Newtonianphysics of collisions. Observers tend to overestimate theimplicit value of C in Bnatural^ collisions and judge simu-lated collisions that are formally inconsistent with the prin-ciple of energy conservation (C > 1) as Bnatural.^ This ev-idence does not fit the assumption that people may haveBinternalized^ a physically plausible value of C Bin orderto reflect knowledge on the natural statistics of our ecolog-ical environment^ (De Sá Teixeira et al., 2014, p. 499). northe assumption that observers are highly sensitive to viola-tions of the principle of energy conservation (Twardy &Bingham, 2002). In particular, the discrepancies betweenour results and those of Twardy and Bingham are worthyof comment.

Twardy and Bingham (2002) presented their participantswith virtual animations depicting a ball falling from high upthat bounced (collided) several times upon the ground at theend of the fall. Theymanipulated the value ofC implied by therebounds of the ball and asked the participants to rate theBnaturalness^ of the motion. The results showed that the nat-uralness ratings sharply decreased when the simulated valueof C was greater than 1—namely, when the animation wasinconsistent with the principle of energy conservation. At least

three sources of information are relevant in the case of a ballbouncing several times upon the ground: the relative height ofthe bounces, their relative period, and the relative bounce ve-locity. These sources are abundant because they repeat them-selves in the sequence of bounces. On the contrary, in the caseof horizontal collisions (our stimuli), only the ratio betweenthe relative pre- and postcollision velocities of the spheres(right part of Eq. 3) provides information about possible vio-lations of the principle of energy conservation. A reason whypeople tend to judge simulated bounces implying C > 1 asBunnatural^ but simulated horizontal collisions as Bnatural^in the same condition might be the greater amount of stimulusinformation available in the former situation. This highlightsthat the outcomes of intuitive physics studies should beinterpreted with caution. Our picture of people’s intuitive un-derstanding of the laws of physics appears to depend on thespecific features of the stimuli used in empirical research (cf.Cooke & Breedin, 1994, on the Bcontext specificity^ of intu-itive physics).10

Author note We thank Veronica Quaini for help in collecting the dataof Experiments 1A and 1B.

Appendix

Themeasures ofPC (physical coefficient of restitution) for theexperimental conditions of Experiments 1A, 1B, and 2 wereobtained using a pendulum device, as is illustrated in Fig. 9.The pendulum spheres are the real spheres referred to in thetext. We used a pendulum because it obeys physical principlessimilar to those involved in horizontal collisions; in particular,the spin of the spheres and the resistance of the medium are

10 In simulated collisions, we cannot exclude that the participants’ sensi-tivity to violations of the principle of kinetic energy might increase withthe realism of the motion of the colliding objects. In this regard, in futurestudies it might be worthwhile to explore the effects of adding rotation tothe colliding objects, and/or the effects of making the colliding objectsmove in response to the application of an external force (e.g., the hit of abilliard cue).

Fig. 9 Three frames of a collision between two pendulum spheres. The symbols α, β, γ, and δ have been added for reference in the text. Arrows havebeen added to indicate which objects are moving at the three stages of the collision event


(ideally) negligible. In the case of pendulum collisions, C canbe determined by this equation:

C ¼h√ 1 − cos δð Þ − √ 1 − cos γð Þ

h i= √ 1 − cos α

�� − √ 1 − cos βð Þ

h i;

ð10Þ

where α, β, γ, and δ denote the angles (in radians) representedin Fig. 9 (for Michottean collisions, β = 0).

For each experimental condition, measures of PC wereobtained by causing collisions between the pendulumspheres and measuring the angles required by the right-hand part of Eq. 10. For instance, in the case of head-oncollisions, we first raised the spheres by hand to certainangles α and β. Then we released the spheres simultaneous-ly and measured the resulting postcollision angles γ and δ.We used the same procedure for the Michottean collisions,except that only one sphere was raised before the collision.Accurate measures of the pre- and postcollision angles wereobtained from videotapes of each collision. We recordedseveral collisions for each experimental condition whilerandomly varying the precollision angles α and β. Amongthe recorded collisions, we chose ten Bvalid^ collisions thathad the least lateral and/or rotational sphere motion. Wecomputed C for each of the ten collisions using Eq. 10,and the PCs were obtained by averaging the ten values ofC for each experimental condition.

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